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In this work systems of sines 1∪ {sin (n − β)t}n≥1 and 1∪ {cos (n − β)t}n≥1 cosines are considered, where β is real parameter. Criterion for the completeness, minimality and basicity of these systems with respect to the parameter β in one subspace of grand-Sobolev space are found.

In the present paper a criterion for basicity of exponential system with linear phase is obtained in Sobolev weight space 1 , (− , ). | | ( ) )

The perturbed systems of sines, which appear when solving some partial differential equations by the Fourier method, are considered in this paper. Basis properties of these systems in weighted Sobolev spaces of functions are studied.

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